Universal Quantum Birthmark: Quantifying the Breakdown of Classical Ergodicity

Ergodicity, associated with chaotic behavior, asserts total memory loss of initial conditions, thus laying the foundation for statistical mechanics. However, in quantum systems, coherence and interference can fundamentally alter this picture. Here we identify a universal quantum birthmark: a persistent memory effect that survives indefinitely even in maximally chaotic, ergodic quantum systems. Employing random matrix theory and group-theoretical arguments, we derive analytical closed-form expressions for the long-time revisitation probability enhancement η of the initial state and its evolutes, always exceeding the classical expectation. We rigorously prove that η=2N/(N+1) for Gaussian Unitary Emsembles and η=3N/(N+2) for Gaussian Orthogonal Emsembles, where N is the dimension of the Hilbert space. Particularly, we show that this enhancement factor originates from fundamental statistical correlations imposed by normalization, not from system-specific dynamics. Furthermore, additional symmetries amplify this birthmark effect, which is quantified in this work. This phenomenon is an important part of the general framework of quantum birthmarks, which also encompasses the revival enhancement of short-time dynamics. Overall, the concept of quantum birthmark introduced here entails an ergodicity breaking present in all closed quantum systems.